hp-FEM for a stabilized three-field formulation of the biharmonic problem

In this paper, a three-field formulation for the biharmonic equation is considered that consists of the potential u, the flux σ and a Lagrange multiplier. The use of stabilization techniques circumvents the discrete inf–sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields. A priori error estimates as well as reliable and efficient a posteriori error estimates are derived. The proof of the latter is based on an implicit, never to be computed, H 2-reconstruction of the discrete potential. Several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p-robust with an efficiency index tending towards the optimal value of one. The numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the Lagrange multiplier for the a priori error estimate is not fulfilled. In particular, h- and h p-adaptivity recover optimal algebraic and exponential converge rates, respectively.